(20) For sets A and B define the set of bijections from A to B B(A,B)...
2. (20 points) Let f: R + R and g: R + R be bijections. Prove that the function G:R2 + R2 defined by G(x, y) = (3f(x) + 4g(y), 2f(x) + 3g(y)) is a bijection.
11. (a) Let A be the open interval (1,5), and let B be the interval (0,8). Define a bijection from A to B (b) Let A = (0,00) and let B = [0,00). Define a bijection from A to B. 12. Is it possible to find two infinite sets A and B such that If your answer is yes, then construct an example 13. Is it possible to find a finite set A such that [AAI = 27?
11. (a)...
5. Recall that if the domain of a function f:B-C is the same as the codomain of a function g: A-B, we can define the composition of these functions fog:A-C given by fºg(a) = f(g(a)). (a) Prove that if f,g: A - A are bijections, then fog: A - A is a bijection. (b) If A is finite with n elements, how many bijections A - A are there? That is, how many elements are in the set Bij(A) :=...
(2) Define the set A by (a) Prove that for any N 20 the set is compact. (b) Prove that for any e>0 there exists some N 2 0 so that for any x A we have (c) Prove that A is totally bounded. d) Prove that A is compact.
Sets A, B, and Care subsets of the universal set U. These sets are defined as follows. U= {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1,6,7,8,9} B = {1, 3, 4, 7, 9) C = {4, 5, 6, 7} Find CU ( BA)'. Write your answer in roster form or as Ø. CU (BNA): = 0 Х 5 ?
(a) Recall that two sets have the same cardinality if there is a bijection between them and that Z is the set of all integers. Give an example of a bijection f: Z+Z which is different from the identity function. (b) For the following sets A prove that A has the same cardinality as the positive integers Z+ i. A= {r eZ+By Z r = y²} ii. A=Z 1.
Problem 8. Given each pair of sets, come up with a formula for a bijection between them You do not need to prove your function is a bijection. Your formula should not be complicated by any means 1. From (0, 1) to (211, 2019) 2. From [0, 1) to (0, 1] 3. From NU (o) to N. 4. From the set of even numbers to 2 5. From the set of odd numbers to Z. 6. r2'2 7. From R...
Question4 please
(1). Let f: Z → Z be given by f(x) = x2. Find F-1(D) where (a) D = {2,4,6,8, 10, 12, 14, 16}. (b) D={-9, -4,0, 16, 25}. (c) D is the set of prime numbers. (d) D = {2k|k Ew} (So D is the set of non-negative integer powers of 2). (2). Suppose that A and B are sets, C is a proper subset of A and F: A + B is a 1-1 function. Show that...
A. (Leftovers from the Proof of the Pigeonhole Principle). As before, let A and B be finite sets with A! 〉 BI 〉 0 and let f : A → B be any function Given a A. let C-A-Va) and let D-B-{ f(a)} PaRT A1. Define g: C -> D by f(x)-g(x). Briefly, if g is not injective, then explain why f is not injective either. Let j : B → { 1, 2, 3, . . . , BI}...
Exercise 3.3.1: Unions and intersections of sets. Define the sets A, B, C, and D as follows: A = {-3, 0, 1, 4, 17} B = {-12, -5, 1, 4, 6} C = {x ∈ Z: x is odd} D = {x ∈ Z: x is positive} For each of the following set expressions, if the corresponding set is finite, express the set using roster notation. Otherwise, indicate that the set is infinite. (c) A ∩ C (d) A ∪...